Question

multiple-choice question
which is the algebraic notation for this translation?
(x - 8, y)
(x - 6, y + 4)

multiple-choice question
we need to know the __ and the __ for rotations.
angle of rotation and the center of rotation
color and the shading

Explanation:

First Question (Translation of \(\boldsymbol{\overline{PQ}}\))

Step1: Analyze \(x\)-coordinate change

For point \(P(3, 0)\) to \(P'(3, 0)\)? Wait, no, wait the original \(P\) and \(P'\)? Wait, looking at the image, original \(P\) is \((3, 0)\) and \(P'\) – wait, maybe I misread. Wait, the translation: let's check \(Q(6, -6)\) to \(Q'(6, -6)\)? No, wait the numbers: Wait, the translation vector? Wait, the options are \((x - 8, y)\) or \((x - 6, y + 4)\)? Wait, no, let's re-express. Wait, original \(P\) is \((3, 0)\), after translation, \(P'\) – wait, maybe the image has a typo, but looking at the options: Wait, the first option is \((x - 8, y)\), second is \((x - 6, y + 4)\)? Wait, no, the user's image shows two options: \((x - 8, y)\) and \((x - 6, y + 4)\)? Wait, no, the first question: Let's take point \(P(3, 0)\) and \(P'\) – wait, maybe the original \(P\) is \((11, 0)\) and \(P'\) is \((3, 0)\)? Wait, \(11 - 8 = 3\), so \(x\)-coordinate change is \(x - 8\), \(y\)-coordinate remains same? Wait, \(Q(14, -6)\) to \(Q(6, -6)\): \(14 - 8 = 6\), \(y\) remains \(-6\). So the translation is \((x - 8, y)\). Wait, but let's confirm:

Original \(P\): let's say \(P(11, 0)\), after translation \(P'(3, 0)\): \(11 - 8 = 3\), \(0\) stays. \(Q(14, -6)\) to \(Q'(6, -6)\): \(14 - 8 = 6\), \(y\) stays. So the algebraic notation is \((x - 8, y)\).

Step2: Confirm with coordinates

For a translation, the rule is \((x + a, y + b)\), where \(a\) is horizontal change, \(b\) vertical. Here, \(x\) decreases by 8 (so \(a = -8\), i.e., \(x - 8\)), \(y\) doesn't change (\(b = 0\)). So the translation is \((x - 8, y)\).

For a rotation, we need two key pieces of information: the angle of rotation (how many degrees we rotate, e.g., \(90^\circ\), \(180^\circ\)) and the center of rotation (the point around which we rotate the figure). This is because a rotation is defined by its center and the angle (and direction, but angle includes direction for positive/negative).

Answer:

\((x - 8, y)\) (assuming the option with \((x - 8, y)\) is correct)

Second Question (Rotation Requirements)